Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(19,23)(21,24);(2,6)(7,8)(10,14)(15,16)(17,18)(19,24)(20,22)(21,23);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,19,20,23)(18,21,22,24);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,22)(21,24);(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|c^2=fg,d^2=f,e^2=gh,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #208472, #208475, #208481, … |
| maximal normal subgroup of | #208472, #208475, #208481, … |
| maximal quotient of | #514220, #514244, #514481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27957);